Mean Value Theorem

#REDIRECT Mean value theorem

Mean value theorem

[[Image:Mvt2.png|frame|right|For any function that is continuous on [''a'', ''b''] and differentiable on (''a'', ''b'') there exists some ''c'' in the interval (''a'', ''b'') such that the secant joining the endpoints of the interval [''a'', ''b''] is parallel to the tangent at ''c''.]] In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the gradient (slope) of the curve is equal to the "average" gradient of the section. It is used to prove theorems that make global conclusions about a function on an interval starting from local hypotheses about derivatives at points of the interval. This theorem was developed by Joseph-Louis Lagrange. Some mathematicians consider this theorem to be ''the'' most important theorem of calculus (see also: the fundamental theorem of calculus). The theorem is not often used to solve mathematical problems; rather, it is more commonly used to mathematical proof other theorems. The mean value theorem can be used to prove Taylor's theorem, of which it is a special case. More precisely, the theorem states: for some Continuous function derivative curve; for every secant, there is some parallel tangent. In addition, the tangent runs through a point located between the intersection points of said secant. :Let ''f'' : [''a'', ''b''] → R be continuous on the closed interval (mathematics) [''a'', ''b''], and derivative on the open interval (''a'', ''b''). Then there exists some ''c'' in (''a'', ''b'') such that ::

f ' (c) = \frac{f(b) - f(a)}{b - a}

It is a generalization of Rolle's theorem, which assumes ''f''(''a'') = ''f''(''b''), so that the right-hand side above is zero. The following reordering is also true and is defined for ''a''=''b'': ::

f (b) - f (a) = f ' (c)(b - a)

Generalization: The theorem is usually stated in the form above, but it is actually valid in a slightly more general setting: We only need to assume that ''f'' : [''a'', ''b''] → R is continuous on [''a'', ''b''], and that for every ''x'' in (''a'', ''b'') the limit

\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}

exists or is equal to � infinity. ==Proof== An understanding of this and the slope will make it clear that the equation of a secant (which intersects (''a'', ''f''(''a'')) and (''b'', ''f''(''b'')) ) is: ''y'' = {[''f''(''b'') - ''f''(''a'')] / [''b'' - ''a'']}(''x'' - ''a'') + ''f''(''a''). The formula ( ''f''(''b'') - ''f''(''a'') ) / (''b'' - ''a'') gives the slope of the line joining the points (''a'', ''f''(''a'')) and (''b'', ''f''(''b'')), which we call a chord of the curve, while ''f'' ' (''x'') gives the slope of the tangent to the curve at the point (''x'', ''f''(''x'') ). Thus the Mean value theorem says that given any chord of a smooth curve, we can find a point lying between the end-points of the chord such that the tangent at that point is parallel to the chord. The following proof illustrates this idea.

Define ''g''(''x'') = ''f''(''x'') + ''rx'', where ''r'' is a constant. Since ''f'' is continuous on [''a'', ''b''] and differentiable on (''a'', ''b''), the same is true of ''g''. We choose ''r'' so that ''g'' satisfies the conditions of Rolle's theorem, which means : $g(a) = g(b) \qquad \Rightarrow \qquad f(a) + ra = f(b) + rb$ : $\Rightarrow \qquad r = - \frac{ f(b) - f(a) }{ b - a}$ By Rolle's Theorem, there is some ''c'' in (''a'', ''b'') for which ''g'' '(''c'') = 0, and it follows : $f ' (c) = g ' (c) - r = 0 - r = \frac{ f(b) - f(a) }{ b - a}$ as required.

==Cauchy's mean value theorem== Cauchy's mean value is the more generalised form of mean value theorem. It states: If functions ''f''(''t'') and ''g''(''t'') are both continuous on the closed interval [''a'', ''b''] and differentiable on the open interval (''a'', ''b''), then there exists some ''c'' in (''a'', ''b''), such that :

\frac {f'(c)} {g'(c)} = \frac {f(b) - f(a)} {g(b) - g(a)}.

Cauchy's mean value theorem can be used to prove l'Hopital's rule. The mean value theorem is the special case of Cauchy's mean value when ''g''(''t'') = ''t''.

===Proof of Cauchy's mean value theorem=== The proof of Cauchy's mean value theorem is based on the same idea as the proof of mean value theorem. We aim to transform the curve defined by y = y(t) and x = x(t), so that it satisfies the conditions of Rolle's theorem. We define a new function: : $F(t) = y(t) - m x(t)$ where ''m'' is a constant, so that : $F(a) = F(b) \qquad \Rightarrow \qquad m = \frac {y(b) - y(a)} {x(b) - x(a)}$ Since ''F'' is continuous and ''F''(''a'') = ''F''(''b''), by Rolle's theorem, there exists some ''c'' in (''a'', ''b'') such that ''F''′(''c'') = 0, i.e. : $F'(c) = 0 \ = \ y'(c) - \frac {y(b) - y(a)} {x(b) -x(a)} x'(c)$ : $\Rightarrow \qquad \frac {y'(c)} {x'(c)}\ = \ \frac {y(b) - y(a)} {x(b) - x(a)}$ as required.

== Mean value theorems for integration == The first mean value theorem for integration states: :If ''f'' : [''a'', ''b''] → R is a continuous function and φ : [''a'', ''b''] → R is an integration positive function, then there exists a number ''x'' in (''a'', ''b'') such that ::

\int_a^b f(t)\varphi (t) \, dt  \quad = \quad  f(x) \int_a^b \varphi (t) \, dt.

In particular (φ(''t'') = 1), there exists ''x'' in (''a'', ''b'') with :

\int_a^b f(t) \, dt  \quad = \quad f(x) (b - a).

The second mean value theorem for integration states: :If ''f'' : [''a'', ''b''] → R is a positive and monotone decreasing function and φ : [''a'', ''b''] → R is an integrable function, then there exists a number ''x'' in (''a'', ''b''] such that ::

\int_a^b f(t) \varphi (t) \, dt  \quad =  \quad ( \lim_{t \to a} f(t) ) 
\cdot \int_a^x \varphi (t) \, dt.

== See also == * arithmetic mean * Newmark-beta method == External links == * [http://mathworld.wolfram.com/Mean-ValueTheorem.html Mathworld: Mean-Value Theorem] Calculus Theorems th:ทฤษฎีบทค่ากลาง

Mean value theorem

This mvt things badly needs a picture to clear things up. The agreed policy is that all words in a species' official common name should be capitalised, other than those following a hyphen if they refer to a part of the animal: "Bald Eagle", "Red-necked Phalarope", "Wilson's Storm-Petrel". The biology convention appears to be applicable to math as well. User:Pizza Puzzle :...and were the mean value theorem an animal, I'd agree with you. -- User:The Anome 14:13 2 Jul 2003 (UTC) Well, so much for logic...do you have some conventional ruling which you feel somehow overrids that and the convention that proper names should be capitalized? How is it easier to read Joseph-Louis LaGrange; instead of simply LaGrange. Students of the Mean Value Thoerem do not need to be on a first name basis with Joe. User:Pizza Puzzle Shouldn't the theorem text in the image be cropped? The theorem is stated in the article, and in greater detail. User:Dysprosia 11:42, 29 Aug 2003 (UTC) ---- The illustration says "there exist", where evidently it means "there exists". Could someone correct this? -- I don't know what software created the illustration. Thanks. User:Michael Hardy 00:22, 15 Nov 2003 (UTC) == Calculus table move == I moved the calculus table down to the See also section. Both the table and diagram wouldn't fit side by side at many window sizes, and there wasn't room for it right under the main picture or in any other section. Besides this, it makes sense in see also — it is, after all, a list of related links. User:Dcoetzee 02:33, 28 Apr 2005 (UTC)

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